Journal of Mathematical Chemistry

, Volume 49, Issue 8, pp 1587–1598 | Cite as

New Nordhaus-Gaddum-type results for the Kirchhoff index

  • Yujun Yang
  • Heping Zhang
  • Douglas J. Klein
Original Paper


Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index is the sum of resistance distances between all pairs of vertices in G. Zhou and Trinajstić (Chem Phys Lett 455(1–3):120–123, 2008) obtained a Nordhaus-Gaddum-type result for the Kirchhoff index by obtaining lower and upper bounds for the sum of the Kirchhoff index of a graph and its complement. In this paper, by making use of the Cauchy-Schwarz inequality, spectral graph theory and Foster’s formula, we give better lower and upper bounds. In particular, the lower bound turns out to be tight. Furthermore, we establish lower and upper bounds on the product of the Kirchhoff index of a graph and its complement.


Resistance distance Kirchhoff index Nordhaus-Gaddum-type result 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wiener H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRefGoogle Scholar
  2. 2.
    Bavelas A.: A mathematical model for small group structures. Hum. Organiz. 7(3), 16–30 (1948)Google Scholar
  3. 3.
    Klein D.J., Randić M.: Resistance distance. J. Math. Chem. 12, 81–95 (1993)CrossRefGoogle Scholar
  4. 4.
    Zhu H.-Y., Klein D.J., Lukovits I.: Extensions of the Wiener number. J. Chem. Inf. Comput. Sci. 36, 420–428 (1996)Google Scholar
  5. 5.
    Gutman I., Mohar B.: The Quasi-Wiener and the Kirchhoff indices coincide. J. Chem. Inf. Comput. Sci. 36, 982–985 (1996)Google Scholar
  6. 6.
    Estrada E., Hatano N.: Topological atomic displacements, Kirchhoff and Wiener indices of molecules. Chem. Phys. Lett. 486, 166–170 (2010)CrossRefGoogle Scholar
  7. 7.
    Xiao W.J., Gutman I.: Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110, 284–289 (2003)Google Scholar
  8. 8.
    Ivanciuc O., Klein D.J.: Building-block computation of Wiener-type indices for the virtual screening of combinatorial libraries. Croat. Chem. Acta 75, 577–601 (2002)Google Scholar
  9. 9.
    Ivanciuc O., Klein D.J.: Computing Wiener-type indices for virtual libraries generated from heteroatom-containing building blcoks. J. Chem. Inf. Comput. Sci. 42, 8–22 (2002)Google Scholar
  10. 10.
    Klein D.J.: Graph geometry, graph metrics and Wiener. MATCH Commun. Math. Comput. Chem. 35, 7–27 (1997)Google Scholar
  11. 11.
    Klein D.J.: Resistance-distance sum rules. Croat. Chem. Acta 75, 633–649 (2002)Google Scholar
  12. 12.
    Klein D.J., Dos̆lić T., Bonchev D.: Vertex-weightings for distance moments and thorny graphs. Discrete Appl. Math. 155, 2294–2303 (2007)CrossRefGoogle Scholar
  13. 13.
    Klein D.J., Lukovits I., Gutman I.: On the definition of the hyper-wiener index for cycle-containing structures. J. Chem. Inf. Comput. Sci. 35, 50–52 (1995)Google Scholar
  14. 14.
    Palacios J.L.: Closed-form formulas for Kirchhoff index. Int. J. Quantum Chem. 81, 135–140 (2001)CrossRefGoogle Scholar
  15. 15.
    Palacios J.L.: Foster’s formulas via probability and the Kirchhoff index. Method Comput. Appl. Prob. 6, 381–387 (2004)CrossRefGoogle Scholar
  16. 16.
    Yang Y.J., Jiang X.Y.: Unicyclic graphs with extremal Kirchhof index. MATCH Commun. Math. Comput. Chem. 60, 107–120 (2008)Google Scholar
  17. 17.
    Zhang H.P., Jiang X.Y., Yang Y.J.: Bicyclic graphs with extremal Kirchhoff index. MATCH Commun. Math. Comput. Chem. 61, 697–712 (2009)Google Scholar
  18. 18.
    Zhang H.P., Yang Y.J.: Resistance distance and Kirchhoff index in circulant graphs. Int. J. Quantum Chem. 107, 330–339 (2007)CrossRefGoogle Scholar
  19. 19.
    Zhang H.P., Yang Y.J., Li C.W.: Kirchhoff index of composite graphs. Discrete Appl. Math. 107, 2918–2927 (2009)CrossRefGoogle Scholar
  20. 20.
    Zhang W., Deng H.Y.: The second maximal and minimal Kirchhoff indices of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 61, 683–695 (2009)Google Scholar
  21. 21.
    Zhou B., Trinajestić N.: A note on Kirchhoff index. Chem. Phys. Lett. 455(1-3), 120–123 (2008)CrossRefGoogle Scholar
  22. 22.
    Zhou B., Trinajestić N.: On resistance-distance and Kirchhoff index. J. Math. Chem. 46(1), 283–289 (2009)CrossRefGoogle Scholar
  23. 23.
    Zhou B., Trinajestić N.: The Kirchhoff index and the matching number. Int. J. Quantum Chem. 109(13), 2978–2981 (2009)CrossRefGoogle Scholar
  24. 24.
    Nordhaus E.A., Gaddum J.W.: On complementary graphs. Am. Math. Monthly 63, 175–177 (1956)CrossRefGoogle Scholar
  25. 25.
    Alavi Y., Behzard M.: Complementary graphs and edge chromatic numbers. SIAM J. Appl. Math. 20, 161–163 (1971)CrossRefGoogle Scholar
  26. 26.
    Chartrand G., Schuster S.: On the independence numbers of complementary graphs. Trans. New York Acad. Sci. Ser. II 36, 247–251 (1974)Google Scholar
  27. 27.
    Goddard W., Henning M.A.: Nordhaus-Gaddum bounds for independent domination. Discrete Math. 268, 299–302 (2003)CrossRefGoogle Scholar
  28. 28.
    Hong Y., Shu J.: A sharp upper bound for the spectral radius of the Nordhas-Gaddum type. Discrete Math. 211, 229–232 (2000)CrossRefGoogle Scholar
  29. 29.
    Liu H., Lu M., Tian F.: On the ordering of trees with the general Randić index of the Nordhaus-Gaddum type. MATCH Commun. Math. Comput. Chem. 55, 419–426 (2006)Google Scholar
  30. 30.
    Zhang L., Wu B.: The Nordhaus-Goddum-type inequalities for some chemical indices. MATCH Commun. Math. Comput. Chem. 54(1), 189–194 (2005)Google Scholar
  31. 31.
    Zhou B., Gutman I.: Nordhaus-Gaddum-type relations for the energy and Laplacian energy of graphs. Bull. Cl. Sci. Math. Nat. Sci. Math. 134, 1–11 (2007)Google Scholar
  32. 32.
    Zhou B.: On sum of powers of the Laplacian eigenvalues of graphs. Linear Algebra Appl. 429, 2239–2246 (2008)CrossRefGoogle Scholar
  33. 33.
    Cameron P.J.: Strongly regular graphs. In: Beineke, L.W., Wilson, R.J. (eds) Selected Topics in Graph Theory, pp. 337–360. Academic Press, London (1979)Google Scholar
  34. 34.
    Godsil C., Royle G.: Algebric Graph Theory. Springer, New York (2001)Google Scholar
  35. 35.
    van Lint J.H., Wilson R.M.: A Course in Combinatorics. Cambridge University Press, New York (1992)Google Scholar
  36. 36.
    Cvetkovic D., Doob M., Sachs H.: Spectra of Graphs: Theory and Application. Academic Press, New York (1980)Google Scholar
  37. 37.
    Foster R.M.: The average impedance of an electrical network. In: Edwards, J.W. (ed.) Contributions to Applied Mechanics, pp. 333–340. Ann Arbor, Michigan (1949)Google Scholar
  38. 38.
    Anderson W.N., Morley T.D.: Eigenvalues of the Laplacian of a graph. Lin. Multilin. Algebra 18, 141–145 (1985)CrossRefGoogle Scholar
  39. 39.
    Bondy J.A., Murty U.S.R.: Graph Theory with Applications. North Holland, Amsterdam (1976)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceYantai UniversityYantaiPeople’s Republic of China
  2. 2.Mathematical Chemistry GroupTexas A&M University at GalvestonGalvestonUSA
  3. 3.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

Personalised recommendations